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According to Cameron, the hypergame paradox proceeds as follows: A game is considered as well founded if ANY play of the game ends in a finite number of moves. A hypergame is where the first player chooses well founded a game, and then the second player begins that game (the game is played with the roles reversed). Is the hypergame wf? It should be, because the first player has to choose a wf game, and so it must end in a finite number of moves. But then, as it is well founded, the second player can choose the hypergame, and then the players can choose the hypergame ad infinitum, contradicting the hypergames wf-ness. I do not see how this contradicts the hypergames wf-ness. The definition of a hypergame clearly sttaes that ANY play of the game ends in a finite number of moves. Well, the hypergame begins on the first go, and then if the second player chose tic-tac-toe (or any wf game except the hypergame), the game would be over in a finite number of moves in this case, making it well founded. Surely one case of the hypergame not ending in a finite number of moves does not contradict its wf-ness. I feel like I am being incredibly stupid but I just can't see how this is a paradox.

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When saying "according to ..." it is common to give an accurate citation and reference. – Asaf Karagila Dec 17 '12 at 11:54

" Surely one case of the hypergame not ending in a finite number of moves does not contradict its wf-ness." It does : The definition of well-foundedness is exactly this : "there is no case of the game that doesn't end in a finite number of moves" (there is no way to play the game for infinitely many moves)

the "any play of the game is ..." phrase means that for every way to play the game, the game ends in a finite number of moves ; it doesn't mean that there exists one particular way to play the game that ends in a finite number of moves. You showed that there is one way that ends in a finite number of moves (the player chooses tic-tac-toe), and you also showed that there is one way that never ends (the player chooses the hypergame at every turn). This last one is enough to contradict well-foundedness.

You should try to avoid the word "any" because its meaning completely changes from sentence to sentence : "if there is any $x$ such that $P(x)$, then ..." is diametrically different from "if any $x$ satisfies $P(x)$, then ...".

For any set $S$ of well-founded games, there is a corresponding hypergame $G_S$. $G_S$ is a well-founded game, and $G_S \notin S$ (otherwise, you deduce that you can play it indefinitely, so it cannot be well-founded). So for any set of well-founded games, you just made a new well-founded game not in this set.

So the paradox is resolved by observing that there is no set of all well-founded games, a bit like how there is no set of all sets.

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If a statement says that something happens for ANY play of the game, then coming up with a single counterexample is enough to contradict that statement. For instance, if I say that "all real numbers have multiplicative inverses" then you should immediately say that this is false, because we know that 0 is an example of a real number that does not have a multiplicative inverse, so my statement is false.

Here, the statement is "all instances of hypergame end in a finite number of turns". This is false, because you exhibited a specific example where hypergame goes on for an infinite number of turns.

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Is the hypergame a "game" at all? It certainly looks like one at first sight, but in order for it to be well-defined, its rules needs to specify precisely what a "game" is such that we can know for sure what the first player's legal moves even are. And can the hypergame then be a "game" according to its own rules?

If a "game" is something that can be formalized in ordinary set theory, then I think it is difficult to make the hypergame a first-class "game" itself. Consider: does the term" game" imply any limit to how many moves a player has to choose from at any time. If there is one (for example that there can only be countably many moves to choose from), then the total number of possible games is larger than that, so the hypergame is not itself a game. On the other hand, if a player can have arbitrarily many moves to chose between, then there are so many possible games that they do not form a set, and again the hypergame cannot be a game.

True enough, these set-theoretical technicalities may seem like avoiding the issue -- who says that games are to be sets at all? I know of several different attempts to give games or game-like concepts a role that is more fundamental than set theory in the development of mathematics. However, if you want to present reasoning about a pre-set game-based formalism, then the onus is on you to make sure that your formalism doesn't allow paradoxical reasoning -- just as set theories need to avoid Russell's paradox somehow, then game theories need to avoid the hypergame paradox somehow.

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